This is not really going to be an answer to your question but here is a long comment. It's a little more conceptual to rewrite the defining relations of $A$ in the form
$$A = F[x_1, \dots x_n]/(p(t) = \prod (t - x_i))$$
where $t$ is not an element of $A$ but shorthand for saying that we are imposing exactly the relations implied by the polynomial identity $p(t) = \prod (t - x_i)$, which has the effect of "universally splitting" $p$. Our relations imply that $p(x_1) = 0$, hence that the subalgebra generated by $x_1$ is $A_1 = F[x_1]/p(x_1)$, which has dimension $n$. Working over this subalgebra, we can divide $p(t)$ by $t - x_1$, giving a polynomial identity of the form
$$\frac{p(t)}{t - x_1} = \prod_{i \neq 1} (t - x_i)$$
where the LHS and hence the RHS lives in $A_1[t]$. Writing $p_1(t)$ for the LHS, this gives $p_1(x_2) = 0$, hence the subalgebra generated by $x_2$ over $A_1$ is (possibly a quotient of) $A_2 = A_1[x_2]/p_1(x_2)$, which has dimension at most $n-1$.
Continuing in this way we find that $A$ is spanned as a vector space by the monomials $x_1^{i_1} x_2^{i_2} \dots x_n^{i_n}$ where $0 \le i_j \le j-1$, and so has dimension at most $n!$.
Now let $L$ be the splitting field generated inside $\overline{F}$ by the roots $\theta_i$, and consider the extension of scalars $A_L = A \otimes_F L$. By construction there are exactly $n!$ $L$-homomorphisms $A_L \to L$, corresponding to bijections between the $x_i$ and the $\theta_i$, each of which has kernel a distinct maximal ideal $(x_i - \theta_{\sigma(i)})$. This gives a homomorphism
$$A_L \to \prod_{\sigma \in S_n} L$$
which is surjective by the Chinese remainder theorem, and since $\dim_L A_L \le n!$ it must be an isomorphism. So $\dim_F A = n!$ and $A_L$ is isomorphic to a product
$$\boxed{ A_L \cong \prod_{\sigma \in S_n} L }$$
of $n!$ copies of $L$.
To recover $A$ from $A_L$ we apply Galois descent: $A$ is the fixed subalgebra of $A_L$ under the action of the Galois group $G = \text{Gal}(L/F)$ on $A_L$ given by its action on the right tensor factor $L$ of $A \otimes_F L$. Thinking of $G$ as a subgroup of $S_n$ acting on the $\theta_i$ by permutation, if $g \in G$ sends $\theta_i$ to $\theta_{g(i)}$ then it sends the maximal ideal $(x_i - \theta_{\sigma(i)})$ to $(x_i - \theta_{g(\sigma(i))})$ (possibly I need to insert some inverses here but I am going to blithely ignore this because it doesn't affect the substance of the argument). So $G$ acts by permuting the maximal ideals. The orbits of this action can be identified (non-canonically) with the $\frac{n!}{|G|}$ cosets of $G$ inside $S_n$, and taking fixed points has the effect of collapsing each orbit into a single copy of $L$; the result is that $A$ is isomorphic to a product
$$\boxed{ A \cong \prod_{\sigma \in S_n/G} L }$$
of $\frac{n!}{|G|}$ copies of $L$. This is of course compatible with the observation that by construction there are $n!$ $F$-homomorphisms $A \to L$, which have the same kernel if they are related by the action of $G$. The above argument is a somewhat indirect way to show that the kernels of these homomorphisms account for all maximal ideals of $A$ and that their intersection is zero.
Now for the bad news: this doesn't really help us explicitly compute the maximal ideals of $A$. It's already clear that these are exactly the kernels of the homomorphisms $A \to L$ (this follows e.g. from the Nullstellensatz), so we just get that the elements of these ideals are polynomial identities satisfied by the roots $\theta_i$, and I don't know any general results about how to compute these. Consider for example the case $F = \mathbb{Q}, p = \Phi_n$ of the cyclotomic polynomials; here we can take the $\theta_i$ to be the primitive $n^{th}$ roots of unity in $\mathbb{C}$, which satisfy many "incidental" relations, and I don't see any obvious way to deduce these relations directly from $p$ or the Galois group. A similar example where there will be many "incidental" relations occurs if we take $F$ to be a finite field; generally there will be more incidental relations the smaller the Galois group $G$ (and hence the splitting field $L$) is.
I guess one potential approach is to consider the increasing sequence $L_i = F[\theta_1, \dots \theta_i] \subseteq L$ and try to compute the corresponding sequence of minimal polynomials of $\theta_i$ over $L_{i-1}$. I don't know how easy this is to work or compute with in practice, though.
